The endomorphism kernel property for modular \(p\)-algebras and Stone lattices of order \(n\) (Q2900198)

In [Commun. Algebra 32, No. 6, 2225--2242 (2004; Zbl 1060.06018)], \textit{T. S. Blyth} et al. introduced the concept of endomorphism kernel property (EKP for short) as follows: an algebra \(A\) has EKP if every nontrivial congruence on \(A\) is the kernel of some endomorphism on \(A\). They also proved that finite Boolean algebras and finite chains possess EKP and a finite distributive lattice has EKP iff it is a product of chains. They got a full characterization of finite De Morgan algebras having EKP. For finite Stone algebras, EKP was studied by \textit{H. Gaitán} and \textit{Y. J. Cortés} in [JP J. Algebra Number Theory Appl. 14, No. 1, 51--64 (2009; Zbl 1191.06007)]. NEWLINENEWLINENEWLINE The author uses the so-called triple construction for Stone algebras and modular \(p\)-algebras (developed by T.Katriňák) for proving that finite Stone algebras which are Stone lattices of order \(n\) possess EKP. He characterizes modular \(p\)-algebras having EKP and finite distributive \(p\)-algebras with this property. He proves that a finite distributive lattice \(L\) having the constant 1 among its base operations has EKP iff \(L\) is a chain.